nLab derived microlocalization

Idea

Microlocalization is a tool to study the propagation of singularities of solutions of partial differential systems, in order to study pullbacks of solutions of differential systems (which generalize products of distributions) and prove index theorems.

Derived microlocalization is an adaptation of the theory of microlocalization to the setting of derived global analytic geometry.

Motivation for a derived notion of microlocalization

In non-smooth situations, the usual normal and conormal bundle used in classical microlocalization is not well behaved, and one needs to take derived versions of them. Moreover, to study global analytic index theory, one needs a formulation of the theory of microlocalization in terms of derived loop stacks, in order to work out a cyclic and Hochshild version of index theorems (similar to the one developed by Kashiwara and Schapira in the microlocal formulation of index theory) that works on an arbitrary (e.g. integral) basis.

Loop space and deformation to the normal bundle: method I

We would like to define a family over (say) the disc $D^1$ whose fiber at $0$ is $LX:=\Hom(S^1,X)$ and whose fiber at $1$ is $[LX/S^1]$. In the case of a usual scheme, this is done (by Vezzosi) by showing that $LX$ identifies with $\Hom(BG_a,X)$, and by making $G_a$ act multiplicatively on the action of $BG_a$ on $LX$, to get the trivial action at $0$ and the usual action at $1$. This gives the desired family. Recall that $BG_a$ is the affinization of $B\mathbb{Z}\cong S^1$.

Loop space and deformation to the normal bundle: method II

Microlocalization in derived geometry involves the proper definition of the deformation to the normal bundle of a closed embedding $Y\subset X$ of global analytic spaces (or even stacks). One needs to consider a derived loop space approach to this construction because it allows to avoid the use of denominators in the definition of the Chern character (following Connes-Loday-Toen-Vezzosi) and in the development of more general global analytic index theory with integral coefficients.

The space of paths on $X$ based on $Y$ is the groupoid acting on $Y$ given by

$$P_Y X:=\{f\in \Hom(\Delta^1,X),\;f(0)\in Y,\;f(1)\in Y\}.$$

More concretely, this is given by the homotopy pullback

$$\array{ P_Y X &\to& \Hom(\Delta^1,X) \\ \downarrow && \downarrow^{\mathrlap{ev_0\times ev_1}} \\ Y\times Y &\to& X\times X }$$

Notice that the natural projection $P_Y X\to Y\times Y$ makes $P_Y X$ a groupoid (of paths in $X$) acting on $Y$. In the case of the diagonal immersion $Y=M\hookrightarrow M\times M=X$, we get

$$P_Y X\cong LM:=\Hom(S^1,M).$$

There is a natural projection $p:P_Y X\to \Hom(\Delta^1,X)\sim X$ and $P_Y X$ is equiped with the natural structure of a groupoid acting on $Y$ through the projection $P_Y X\to Y\times Y$. A more explicit description of the loop space (that is obtained by using the homotopy $\Delta^1\sim \Delta^0$) is given by the homotopy pullback

$$P_Y X=Y\times^h_X Y,$$

which clearly has an interesting meaning only in the setting of derived geometry. See also at derived loop space and at Hochschild cohomology.

To make the above construction work for general Artin stacks, one needs to make it local for the smooth topology. This is done by replacing the loop space groupoid $P_Y X$ acting on $Y$ by its formal completion along the identity morphism. This gives a formal groupoid $\hat{P}_Y X$ that is equivalent modulo the HKR theorem (in characteristic $0$) to $T_Y[-1]X$ for a general Artin stack. Similarly, we will have $\hat{L}M:=\hat{P}_M(M\times M)\cong T[-1]M$ by HKR.

The deformation to the normal bundle in strict derived global analytic geometry is then simply given by the (false to be corrected) formula (with $D^1=\mathbb{M}(R\{X\}^\dagger)$ for $R$ the base ind-Banach ring)

$$\widetilde{P_Y X}:=\{f\in \Hom_{D^1}(\Delta^1\times D^1,X\times D^1),\;f(0,0)\in Y,\; f(1,0)\in Y,\;f(-,1)\in (X\backslash Y)\Rightarrow f(x,t)\in (X\backslash Y)\forall t\neq 0\}.$$

More concretely, this is given by the homotopy pullback (where $U(1)=\mathbb{M}(R\{X,Y\}^\dagger/(XY-1))$ and we use the homotopy equivalence $\Hom(\Delta^1,X)\sim X$)

$$\array{ \widetilde{P_Y X} &\to& \Hom_{D^1}(\Delta^1\times D^1,X\times D^1) \\ \downarrow && \downarrow^{\mathrlap{ev_{(0,0)}\times ev_{(1,0)}\times \ev_{(-,U(1))}\times \ev_{(-,1)}}} \\ Y\times Y\times \Hom(\Delta^1\times U(1),(X\backslash Y))\times (X\backslash Y) &\stackrel{i\times j\times k}{\to}& X\times X\times \Hom(\Delta^1\times U(1),X) \times X}$$

It has an evident natural projection $t:\widetilde{P_Y X}\to D^1$, and a natural projection $\widetilde{P_Y X}\to Y\times Y$ that makes it a family of groupoids parametrized by $D^1$ and acting on $Y$. There is also a natural projection $p:\widetilde{P_Y X}\to X$ given by $f\mapsto f(-,1)$. We will denote $s:P_Y X\to \widetilde{P_Y X}$ the fiber $t^{-1}(0)$.

One may complete $\widetilde{P_Y X}$ along the unit of its groupoid structures, to get a family of formal groupoids $\widehat{P_Y X}$ parametrized by $D^1$, whose fiber at $0$ will be the formal loop space $\hat{P}_Y X$ obtained by completing the path groupoid $P_Y X$ acting on $Y$ along its identity morphism, and whose fiber at $1$ will be the formal completion $\hat{X}_Y$ of $X$ along $Y$. There is a natural action of $S^1$ on $\widehat{P_Y X}$. We will still denote $t:\widehat{P_Y X}\to D^1$ the natural projection, $s:\widehat{P}_Y X\to \widehat{P_Y X}$ the fiber $t^{-1}(0)$, and $p:\widehat{P_Y X}\to X$ the natural projection (evaluation at $(-,1)$).

In the particular case of the diagonal embedding $Y=M\hookrightarrow M\times M=X$ over a field of characteristic $0$, the quotient of $X$ by this family of formal groupoids gives back (modulo a convenient HKR theorem) exactly Simpson’s non-abelian Hodge structure, that gives a family of formal stacks over $D^1$ whose fiber at $0$ is the tangent space $TX$ (seen as the quotient of $X$ by the trivial action of $T[-1]X$) and whose fiber at $1$ is the so-called de Rham space $X_{dR}$ of $X$.

A derived analog of microlocalization

We now work on an arbitrary derived Artin stack $M$, such that the diagonal $Y=M\to M\times M=X$ is an affine closed embedding.

The usual theory of microlocalization may be adapted to the derived global analytic setting by replacing the deformation to the normal bundle $\widetilde{T_Y X}\to \mathbb{A}^1$ by the formal deformation to the normal bundle $\widehat{P_Y X}\to D^1$.

The derived specialization functor is given on $F\in D^b(X)$ by

$$\nu_Y(F):=s^*p^*F\in D^b(\hat{P}_Y X).$$

Remark that the loop space analog $L^*X$ of the odd cotangent bundle $T^*[1]X$ (that should be dual to the odd tangent bundle $T[-1]X$ (related to $LX$ by the HKR theorem) is given by $S^1\otimes X$ (external tensor product by the simplicial circle). We will have for every stack $Y$, the canonical equivalence

$$\Hom(S^1\otimes X,Y)\cong \Hom_{SSets}(S^1,\Hom_{dSt}(X,Y)).$$

The derived Fourier-Sato transformation is given on $F\in D^b(\hat{P}_Y X)$ by

$$\Phi(F):=?.$$

The derived microlocalization functor is given on $F\in D^b(M)$ by

$$\mu(F):=\Phi(\nu_Y(F))$$

for $Y=M\hookrightarrow M\times M=X$.

A derived analytic analog of microlocalization

One may replace the simplicial circle $S^1$, used in the definition of the derived loop space, by the unitary group $U(1)$ of overconvergent global analytic geometry, to get a more analytic theory of microlocalization. In the complex situation, we will have $U(1)\cong S^1$ up to $D^1$-homotopy. Remark that the exponential map $exp(i-):\mathbb{R}\to S^1$ also has a meaning in overconvergent complex analytic geometry, if we see $\mathbb{R}\subset \mathbb{C}$ as a closed subset equipped with its germs of analytic functions.

In the global analytic setting (without imposing homotopy invariance), there is no reason to have $U(1)=D^1\times_{*\coprod *} *$. We thus prefer to use $P^1:=D^1\coprod_{U(1)}D^1$ as a natural global analytic parameter space for paths. We define

$$L^\dagger_Y X:=\{f\in Hom(P^1,X), f(0)\in Y,\; f(\infty)\in Y\}.$$

Remark that up to $D^1_\mathbb{R}$-homotopy, we get

$$L^\dagger_Y X\sim L_Y X.$$

We must now check that the natural morphism

References

• David Ben-Zvi, David Nadler, Loops spaces and connections (arXiv:1002.3636)

• Damien Calaque, Andrei Caldararu? and Junwu Tu?: PBW for an inclusion of Lie algebras (arXiv:1010.0985)

• Damien Calaque, Andrei Caldararu? and Junwu Tu?: On the Lie algebroid of a derived self-intersection (arXiv:1306.5260)